#445859

order	512
nilpotency class	3
isoclinism	isoclinic class of #423438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15824
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,13,17,8)(4,10)(5,9)(6,12)(7,11)(14,23,16,22)(15,21,30,25)(18,24)(19,27)(20,26)(28,31,29,32);(1,3)(2,8)(4,15)(5,16)(6,14)(7,17)(9,23)(10,25)(11,13)(12,22)(18,29)(19,28)(20,30)(21,26)(24,31)(27,32);(2,9,11,12)(3,14,17,16)(10,24,26,27)(15,28,30,29);(1,4,5,18,7,20,6,19)(2,10,12,27,11,26,9,24)(3,15,16,29,17,30,14,28)(8,21,22,31,13,25,23,32)

Polycyclic presentation:

2^9|a^2=efg,c^2=fg,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353055);